Problems

In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)

Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?

Each pair of cities in Wonderland is connected by a flight operated by "Wonderland Airlines". How many cities are there in the country if there are \(105\) different flights? We count a flight from city \(A\) to city \(B\) as the same as city \(B\) to city \(A\) - i.e. the pair \(A\) to \(B\) and \(B\) to \(A\) counts as one flight.

(a) In a regular 10-gon we draw all possible diagonals. How many line segments are drawn? How many diagonals?

(b) Same questions for a regular 100-gon.

(c) Same questions for an arbitrary convex 100-gon.

Draw \(6\) points on a plane and join some of them with edges so that every point is joined with exactly \(4\) other points.

There are \(15\) cities in Wonderland, a foreigner was told that every city is connected with at least seven others by a road. Is this enough information to guarantee that he can travel from any city to any other city by going down one or maybe two roads?

Replace letters with digits to maximize the expression \[NO + MORE + MATH.\] (In this, and all similar problems in the set, same letters stand for identical digits and different letters stand for different digits.)

Jane wrote a number on the board. Then, she looked at it and she noticed it lacks her favourite digit: \(5\). So she wrote \(5\) at the end of it. She then realized the new number is larger than the original one by exactly \(1661\). What is the number written on the board?

Find the biggest 6-digit number such that each of its digits, except for the last two, is equal to the sum of its two right neighbours.

Replace the letters with digits in a way that makes the following sum as big as possible: \[SEND + MORE + MONEY.\]